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Theorem dprdwdOLD 16618
Description: The property of being a finitely supported function in the family  S. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdwd 16612 as of 11-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dprdffOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
dprdffOLD.1  |-  ( ph  ->  G dom DProd  S )
dprdffOLD.2  |-  ( ph  ->  dom  S  =  I )
dprdwdOLD.3  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  ( S `  x
) )
dprdwdOLD.4  |-  ( ph  ->  ( `' ( x  e.  I  |->  A )
" ( _V  \  {  .0.  } ) )  e.  Fin )
Assertion
Ref Expression
dprdwdOLD  |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W
)
Distinct variable groups:    A, h    x, h    x, G    h, i, I, x    .0. , h    ph, x    S, h, i, x
Allowed substitution hints:    ph( h, i)    A( x, i)    G( h, i)    W( x, h, i)    .0. ( x, i)

Proof of Theorem dprdwdOLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dprdwdOLD.3 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  ( S `  x
) )
21ralrimiva 2827 . . 3  |-  ( ph  ->  A. x  e.  I  A  e.  ( S `  x ) )
3 eqid 2452 . . . 4  |-  ( x  e.  I  |->  A )  =  ( x  e.  I  |->  A )
43fnmpt 5640 . . 3  |-  ( A. x  e.  I  A  e.  ( S `  x
)  ->  ( x  e.  I  |->  A )  Fn  I )
52, 4syl 16 . 2  |-  ( ph  ->  ( x  e.  I  |->  A )  Fn  I
)
6 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
73fvmpt2 5885 . . . . . 6  |-  ( ( x  e.  I  /\  A  e.  ( S `  x ) )  -> 
( ( x  e.  I  |->  A ) `  x )  =  A )
86, 1, 7syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  =  A )
98, 1eqeltrd 2540 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  e.  ( S `
 x ) )
109ralrimiva 2827 . . 3  |-  ( ph  ->  A. x  e.  I 
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x ) )
11 nfv 1674 . . . 4  |-  F/ y ( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )
12 nffvmpt1 5802 . . . . 5  |-  F/_ x
( ( x  e.  I  |->  A ) `  y )
1312nfel1 2629 . . . 4  |-  F/ x
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y )
14 fveq2 5794 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  I  |->  A ) `  x
)  =  ( ( x  e.  I  |->  A ) `  y ) )
15 fveq2 5794 . . . . 5  |-  ( x  =  y  ->  ( S `  x )  =  ( S `  y ) )
1614, 15eleq12d 2534 . . . 4  |-  ( x  =  y  ->  (
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )  <-> 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) ) )
1711, 13, 16cbvral 3043 . . 3  |-  ( A. x  e.  I  (
( x  e.  I  |->  A ) `  x
)  e.  ( S `
 x )  <->  A. y  e.  I  ( (
x  e.  I  |->  A ) `  y )  e.  ( S `  y ) )
1810, 17sylib 196 . 2  |-  ( ph  ->  A. y  e.  I 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) )
19 dprdwdOLD.4 . 2  |-  ( ph  ->  ( `' ( x  e.  I  |->  A )
" ( _V  \  {  .0.  } ) )  e.  Fin )
20 dprdffOLD.w . . 3  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
21 dprdffOLD.1 . . 3  |-  ( ph  ->  G dom DProd  S )
22 dprdffOLD.2 . . 3  |-  ( ph  ->  dom  S  =  I )
2320, 21, 22dprdwOLD 16617 . 2  |-  ( ph  ->  ( ( x  e.  I  |->  A )  e.  W  <->  ( ( x  e.  I  |->  A )  Fn  I  /\  A. y  e.  I  (
( x  e.  I  |->  A ) `  y
)  e.  ( S `
 y )  /\  ( `' ( x  e.  I  |->  A ) "
( _V  \  {  .0.  } ) )  e. 
Fin ) ) )
245, 18, 19, 23mpbir3and 1171 1  |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   {crab 2800   _Vcvv 3072    \ cdif 3428   {csn 3980   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4942   dom cdm 4943   "cima 4946    Fn wfn 5516   ` cfv 5521   X_cixp 7368   Fincfn 7415   DProd cdprd 16592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-oprab 6199  df-mpt2 6200  df-ixp 7369  df-dprd 16594
This theorem is referenced by:  dprdfidOLD  16631  dprdfinvOLD  16633  dprdfaddOLD  16634  dmdprdsplitlemOLD  16652  dpjidclOLD  16681
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